On Algorithms for Decomposable Constraints

Abstract

Non-binary constraints are present in many real-world constraint satisfaction problems. Certain classes of these constraints, like the all-different constraint, are “decomposable”. That is, they can be represented by binary constraints on the same set of variables. For example, a non-binary all-different constraint can be decomposed into a clique of binary not-equals constraints. In this paper we make a theoretical analysis of local consistency and search algorithms for decomposable constraints. First, we prove a new lower bound for the worst-case time complexity of arc consistency on binary not-equals constraints. We show that the complexity is O(e), where e is the number of constraints, instead of O(ed), with d being the domain size, as previously known. Then, we compare theoretically local consistency and search algorithms that operate on the non-binary representation of decomposable constraints to their counterparts for the binary decomposition. We also extend previous results on arc consistency algorithms to the case of singleton arc consistency.